Spectral gap conjecture

In ergodic theory, a branch of mathematics, the spectral gap conjecture of Alexander Lubotzky, Ralph S. Phillips, and Peter Sarnak is a statement on the spectral gaps of certain actions of a free group on the sphere $S^2$.

Statement

Any matrix $U\in SU(2)$ defines an isometry of the sphere $S^2$, which in turn defines an operator ${\varphi }_U$ on the Hilbert space $L^2(SU(2))$. The spectral gap conjecture states that for any integer $n>2$, if $n$ isometries $U_1,\dots ,U_n$ are chosen uniformly at random, then the operator ${\varphi }_{U_1}+{\varphi }_{U_1}^{-1}+\cdots +{\varphi }_{U_n}+{\varphi }_{U_n}^{-1}$ has a nontrivial spectral gap with probability 1.

Progress

In 2007, Jean Bourgain and Alex Gamburd proved that when the matrices $U_i$ have entries which are all algebraic numbers up to simultaneous conjugation, the resulting operator has a spectral gap. This result was later generalized to the case of $SU(d)$. It is known that either there is a nontrivial spectral gap with probability 1 or that the spectral gap is trivial with probability 1. If true, the statement would have applications to quantum computing and the design of universal quantum gate sets.